Library 


UNIVERSITY  OF  CALIFORNIA 

DEPARTMENT  OF  CIVIL  ENGINEERING 

BERKELEY.  CALIFORNIA 


CONCRETE     COMPUTATION 
CHARTS 
BY 

RICHARD     T.      DANA 

Mem.  Am.   Soc.   C.   £. 
Mem.   A.   I.   M.  &  M.   E. 
Mem,  Yale  Eng.   Assn. 
Ch.  Eng.,  .Construction  Service  Company 

AND 
JAMES     M.      KINGSLEY,      B.    S. 


NEW  YORK 
1922 


Engineering 
Library 


Copyright,   1922, 
by 

R.   T.  DANA 


FOREWORD 

1.  The  purpose  of  this  Book  of  Charts  is  to  fur- 
nish to  the  Designing  Engineer,  the  Draftsman,  the  Es- 
timator,and  the  Student  a  concise  and  complete  appara- 
tus for  Proportioning  the  Parts  and  Estimating  the  Cost 
of  Concrete  Structures. 

2.  The  formulas  for  Moments  of  Resistance  were 
derived  in  1904  and  are  similar  in  effect  to  those  sug- 
gested by  the  American  Society  of  Civil  Engineers  in 
1917.  They  are  here  given  algebraically  and  are  plotted 
on  charts  for  convenience  in  computation.  In  this  form 
they  can  be  used  to  determine  the  sizes,  shapes,  compo- 
sition and  cost  of  Concrete  beams,  slabs,  columns,  etc. 
with  the  minimum  expenditure  of  time  and  risk  of  error. 

3.  Careful  directions  accompany  these  oharts,  by 
the  aid  of  which  the  necessary  quantities  may  be  read 
directly,  thus  avoiding  the  necessity  of  arithmetical  or 
algebraic  calculation,  minimizing  the  risks  of  error  al- 
ways present  in  such  calculations,  and  greatly  expediting 
the  process  of  design.  The  use  of  these  charts  does  not 
require  a  knowledge  of  the  rather  complex  mechanical  the- 
ory of  flexure  in  Reinforced  Concrete  Beams. 


TABLE  OF  CONTENTS 


Example  : 


Floor  System 

Beam  Design 


Description: 


Charts  Or  -  1  to  4 

Chart  Cr  -  5 

Charts  Cr  -  6  to  10 

Chart  Cr  -  11 


Charts  Cr  -  12  to  17 

Notes  on  theory,  method  of  computation  and  application  to  field  work 
Classification  of  Concrete  for  Various  Uses 
Tests  for  Voids 

Description:  Charta  Cr  -  18  to  19 
Example:  Cost  of  1  cu.  yd.  of  1:3:6  concrete 
Formulas  for  reinforced  concrete  construction 
Standard  Notation 

Rectangular  beams 

T-beams 

Beams  reinforced  for  compression 

Shear,  bond  and  web  reinforcement 

Columns 
Formulas 

Rectangular  beams 

Columns 


Charts  for  Bending  Moments  by  M  »  WL2  ,  WLZ  , 

10 

Charts  for  Effective  Depths  of  R.  C.  Beams  and  Slabs  (Logarithmic) 
Chart  for  Area  of  Steel  Required  for  Beams  and  Slabs 
Chart  for  Effective  Depths  of  R.  C.  Beams  and  Slabs  (Rectilinear) 
Chart  for  Effective  Depths  and  Area  of  Cinder  Concrete  Slabs 
Chart  for  Area  and  Weight  of  Round  Steel  Rods 
Chart  for  Spacing  of  Standard  Round  Rods 
Chart  for  Area  and  Weight  of  Square  Steel  Bars 
Chart  for  Spacing  of  Standard  Square  Bars 
Chart  for  Round  Columns  with  Longitudinal  Reinforcement 
Chart  for  Round  Columns  with  Longitudinal  Reinforcement  and  Hoops 

or  Spirals 

Chart  for  Square  Columns  with  Longitudinal  Reinforcement 
Charts  for  Variables  Required  in  Slab  and  Beam  Formula 

(1)  Values  of  j  depending  on  m  and  n 

(2)  Values  of  k  depending  on  n  and  p 

(1)  Values  of  m  depending  on  n  and  p 

(2)  Values  of  k  depending  on  m  and  n 

Charts  for  Proportions  of  Sand  and  Stone  per  Unit  Volume  of  Cement 
E  -  (Excess  of  cement  and  mortar  )  being  10  and  15% 
E  -  (Excess  of  cement  and  mortar  )  being  20  and  25$ 
Chart  for  Amount  of  Cement  Required  for  Mix  as  Determined  by  Cr  -  15 

and  Cr  -  16 
Chart  for  Compressive  Strengths  of  Concrete  of  Different  Materials 

and  Mixtures 

Chart  for  Cost  per  Cu.  Yd.  of  Materials  for  Concrete  of  Various  Mixes 
Chart  Showing  Building  Code  Requirements  of  26  Cities 
Chart  for  Weights  per  Sq.  Ft.  for  Various  Thicknesses  of  Slabs 


PAGE 
4 
5 
6 
7 
8 
9 

10 
10 
11 
12 
13 
14 
14 
14 
14 
14 
15 
15 
15 
15 
15 
15 

CHARTS 


Cr  -  1,2,3 
Cr  -  4 
Cr  -  5 
Cr  -  6 
Cr  -  7 
Cr  -  8 
Cr  -  8a 
Cr  -  9 
Cr  -  9a 
Cr  -  10 


Cr 
Cr 

Cr 
Cr 
Cr 
Cr 

Cr 
Cr 


11 
12 

13 
13 
14 
14 

15 

16 


Cr  -  17 


Cr 
Cr 
Cr 
Cr 


18 
19 
20 
21 


(2) 


EXAMPLE  :  Design  floor  system  for  room  25  x  47  ft,  with  live  load 
of  80  Ibs.  per  sq.  ft. 

By  using  7  floor  beams  the  area  will  be  divided  into  8  slabs  of 
6  ft.  span. 

Loading  L.  L.  80# 

D.  L.  Floor  60#  (assumed) 

Hanging  Ceiling  1 
Total 


Moment  Enter  chart  Cr  -  1  at  bottom  at  point  indicating  155#. 
Run  up  vertically  to  the  intersection  with  the  inclined  line  indicating 
span  of  6  ft.  and  from  this  intersection  run  horizontally  to  the  right 
or  left  hand  margin  where  the  moment  is  read  as  5550  in.  Ibs.  (Actual- 
ly 5580) 

Slab    The  moment  per  ft.  of  width  is  5550  in.  Ibs.  The  moment 
per  inTof  width  is  463  in.  Ibs.  (5550  -*-12) 

d  -     Enter  Cr  -  4  at  right  hand  margin  at  point  indicating  463  in. 
Ibs.  (.463  thousands  of  in.  Ibs.).  Run  horizontally  to  the  left  to  the 
intersection  with  the  inclined  line  and  from  this  intersection  run  down 
to  the  bottom  of  the  sheet  where  the  affective  depth  is  found  to  tbe 
2.55  in. 

Depth  to  steel  equals  2.55  in.   On  a  thin  slab  such  as  this,  due  to 
difficulty  of  properly  placing  steel,  etc.,  it  is  better  practice  to  make 
the  effective  depth  of  the  slab  somewhat  deeper  -  say  2  3/4  or  3  in. 
With  3/4  in.  of  concrete  below  steel  the  total  thickness  of  slab  will  then 
be  3  1/2  or  3  3/4  in. 

AS  -  Enter  Cr  -  5  at  bottom  at  point  indicating  effective  depth  -  d 
of  2.55  in.  Run  up  vertically  to  intersect  with  inclined  line  and  from 
this  intersection  run  horizontally  to  right  hand  margin  where  As  per  in. 
of  width  is  found  to  be  0.0128  sq.  in.  or  0.154  per  ft.  of  width. 

Now  refer  to  Cr  -  8  and  Cr  -  9  where  the  areas  of  round  and  square 
rods  are  shown.  If  we  select  1/4  in.  square  rods  we  find  the  area  is 
.0625  sq.  in. 

The  spacing  is  found  by  the  formula: 

12  x  (Sectional  area  of  bar  to  be  used) 
Spacing  ••• i ft 

AS 

In  this  case  Spacing    12  x  .0625  _  . ,    * 

-^TO 4.87  in.  Say  4  3/4  in. 

Every  other  bar  should  be  bent  up  at  45°  to  run  over  floor  beams 
at  upper  side  of  slab. 

*  The  same  result  may  be  determined  directly  from  Cr  -  9a. 


(3) 


Beam  Design  The  loading  per  foot  of  beam  is  as  follows: 

L.  L.  80  x  6  *=  480  Ibs. 

D.  L.  Floor  (3  3/4  in.)  47  x  6  —  282 

D.  L.  Ceiling  15  x  6  —  90 
Stem  of  beam  =  100  (assumed) 

Total  952  Ibs. 

Moment        Enter  at  95  at  bottom  of  chart.  Run  up  to  inter- 
section with  line  indicating  26  ft.  span  and  from  this  intersection 
run  horizontally  to  right  hand  margin  where  moment  is  found  to  be 
96,000  in.  Ibs.  for  loading  of  95  Ibs.  For  950  Ibs,  the  moment  would 
be  960,000  in.  Ibs.   (Actually  963300.  Error  «•  0.344$  which  is  insig- 
nificant. ) 

1.  Assume  width  of  stem  of  beam  as  8  in.  Then  total  width  of  T 
-  8-1-2(3.75)  —15.5  in. 

Moment  per  inch  of  width  equals  960,000 -*- 15. 5  "•  6,200  in.  Ibs. 
From  Cr  -  4,d  »  9,35. 

2.  Assume  width  of  stem  of  beam  as  6  in.  Then  total  width  of  T 
equals  6  +  2(3.75)  -13.5  in. 

Moment  per  inch  of  width  equals  960, 000 -r 13. 5  «•  7,130  in.  Ibs. 

From  Cr  -  4,d  =  10  in%  which  gives  a  better  proportioned  and 
more  economic  section. 

AS  -  From  Cr  -  5  it  is  found  that  for  a  depth  of  10  in.,  0,05  sq.  in. 
of  steel  is  required  per  in.  of  width.  Then  for  the  total  width  of  13.5 
in.  0.675  sq.  in.  of  steel  is  required 

Use  4  -  7/16  sq.  bars  Area  «4  x  .19  —0.76  sq.  in.   Two  bars  are 
to  run  at  bottom  of  beam  throughout  and  of  the  other  two,  one  is  to  be 
bent  up  at  45°  at  1/4  points  of  span  and  the  other  at  45°  at  points  1/8 
of  span  from  supports. 


(4) 


WL2 

Cr  -  1  is  for  determining  moments  by  the  formula  M  =*  12   * 
This  is  for  use  Tnrith  the  following  conditions: 

1.  For  floor  slabs  containing  over  two  spans, 

2.  For  beams  of  Interior  spans. 

In  this  formula  W  equals  the  combined  dead  and  livo  load  per  foot; 
L  equals  the  span  length  in  feet. 

In  using  the  diagram,  after  determining  the  combined  dead  and  live 
load,  enter  the  diagram  at  the  bottom  as  indicated  by  the  arrow  marked 
1.,  and  follow  the  vertical  line  representing  the  determined  load  until 
it  intersects  the  inclined  line  indicating  the  proper  span.   Then  fol- 
low the  horizontal  line  from  this  point  of  intersection  over  to  the 
right  or  left  margin  of  the  chart  where  the  moment  in  thousands  of  inch 
pounds  may  be  read. 


C r  •  2  is  for  determining  moments  by  the  formula  M  _  WL  . 
ThisTsTor  use  with  the  following  conditions:         "  10 

1.  For  end  beams  and  spans. 

2.  For  beams  and  slabs  with  their  ends  restrained  and  continuous 
for  two  spans  only. 

In  this  formula  W  equals  the  combined  dead  and  live  load  per  foot; 
L  equals  the  span  length  in  feet. 

In  using  this  diagram  proceed  according  to  directions  for  Cr  -  1. 

WL 
Cr  -  o  is  for  determining  moments  by  the  formula  M  _  — >?—  . 

This  is  for  use  with  the  following  conditions: 
1.   For  simple  beams  and  spans. 

In  this  formula  W  equals  the  combined  dead  and  live  load  per  foot; 
L  equals  the  span  length  in  feet. 

In  using  this  diagram  proceed  according  to  directions  for  Cr  -  1. 


Cr  -  4  is  for  determining  the  effective  depths  of  reinforced  con- 
crete beams  and  slabs. 

1.   For  slabs:   Divide  the  moment  in  inch  pounds  as  determined 
from  Cr  -  1,  Cr  -  2  or  Cr  -  3  by  12.   This  will  give  the  moment  per 
inch  width  of  slab.  Having  determined  this  moment,  enter  the  chart  on 
the  right  hand  margin  where  the  moments  are  indicated  in  thousands  of 
inch  pounds  and,  as  indicated  by  the  arrow  marked  1.,  follow  the  hori- 
zontal line  from  this  point  to  the  intersection  of  the  diagonal  line. 


(5) 


From  the  intersection  with  this  line  follow  the  vertical  line  to  the 
bottom  margin  where  the  effective  depth  (i.e.  the  depth  in  inches  from 
the  top  of  section  to  center  line  of  steel)  is  easily  read. 

2,  For  rectangular  beams:   After  obtaining  the  moment  in  inch 
pounds  from  Cr  -  1,  Cr  -  2  or  Cr  -  3  the  width  of  beam  is  assumed  and 
the  moment  is  divided  by  this  width  thus  giving  the  moment  per  inch  of 
width  of  beam.   After  determining  this  moment  enter  the  chart  on  the 
right  hand  margin  where  the  moments  are  indicated  in  thousands  of  inch 
pounds  and  proceed  as  in  1  above. 

The  width  of  beam  will  be  largely  determined  by  the  lateral  spac- 
ing of  the  parallel  reinforcing  bars  which  should  not  be  less  than  3 
diameters  from  center  to  center.   Nor  should  the  distances  from  the 
side  of  beam  to  the  center  of  the  nearest  bar  be  less  than  2  diameters. 
If  two  layers  of  bars  are  used  the  clear  spacing  between  the  layers  of 
bars  should  be  not  less  than  1  inch.   The  use  of  more  than  tv/o  layers 
is  not  recommended  unless  the  layers  are  tied  together  by  metal  connec- 
tions. As  a  general  rule  the  width  of  the  beams  should  not  be  less  than 
1/3  of  their  depth. 

3.  T-beams:   In  the  design  of  T-beams  the  width  of  the  stem  of  the 
beam  is  assumed  and  the  total  width  of  the  beam  is  taken  as  the  width  of 
the  stem  plus  twice  the  thickness  of  the  flange.   It  should  be  noted 
that  this  width  is  considerably  less  than  that  allowed  by  the  Joint  Com- 
mittee of  the  American  Society  of  Civil  Engineers  which  recommended  that 
the  effective  width  should  be  determined  by  the  following  rules: 

A.  Shall  not  exceed  1/4  of  the  span  length  of  the  beam. 

B.  Its  overhanging  width  on  either  side  of  the  web  shall 
not  exceed  six  times  the  thickness  of  the  slab. 

By  using  the  lesser  width  of  flange  we  find  that  lengthy  and  more  or 
less  difficult  computations  to  provide  against  diagonal  tension  and  shear 
are  obviated,  and  it  is  believed  that  with  the  exception  of  large  struc- 
tures the  greater  simplicity  of  this  method,  together  with  the  additional 
security  of  the  more  conservative  design,  justifies  the  use  of  the  addi- 
tional concrete. 

In  determining  the  depth  D  of  the  steel  the  same  steps  are  taken  as 
in  the  design  of  slabs  and  beams  as  indicated  above. 


Cr  -  5  is  for  use  in  determining  the  area  of  steel  required  for 
slabs  and  beams . 

In  using  the  diagram  enter  the  chart  from  the  bottom  at  the  point 
indicating  the  effective  depth  as  determined  from  Cr  -  4  or  Cr  -  6  and 
follow  the  vertical  line  from  this  point  to  its  intersection  with  the 
diagonal  line.   From  this  intersection  follow  the  horizontal  line  to 
the  right  hand  margin  where  the  area  of  steel  in  square  inches  is  shown. 
This  area  must  be  multiplied  by  the  width  of  the  beam  and  in  the  case  of 
slabs  by  12.   This  will  give  the  total  gross  sectional  area  of  steel  re- 
quired. 

From  Cr  -  8  and  Cr  -  9  the  proper  number  of  rods  to  give  this  area 
may  be  obtained. 


(6) 


In  the  oaae  of  slabs,  Or  -  8a  or  Cr  -  9a  will  give  by  inspection 
the  proper  size,  number  and  spacing  of  rods  or  bars  for  required  area 
of  steel, 


Cr  -  6  is  for  the  same  purpose  as  Cr  -  4,  but  it  is  plotted  on  rec- 
tilinear  paper  thus  giving  a  more  preoise  range  for  slabs  and  beams  of 
depths  of  more  than  6  inches. 


Qr  -  7  is  for  the  design  of  cinder  concrete  slabs. 

1.  On  the  right  hand-  side  it  is  arranged  similarly  to  Cr  -  4  with 
the  exception  that  since  cinder  concrete  as  recommended  by  the  Joint  Com- 
mittee of  the  A.  S.  C.  E.  should  not  be  used  for  reinforced  concrete 
structures  with  floor  slabs  exceeding  8  ft.  span,  the  moments  of  resist- 
ance are  given  with  a  width  of  1  ft.   It  is  therefore  possible  to  enter 
the  diagram  on  the  right  hand  margin  using  the  moment  in  thousands  of  inch 
pounds  as  obtained  from  Cr  -  1,  Cr  -  2  or  Cr  -  3  on  a  vertical  line  and 
from  this  point  run  over  on  the  horizontal  line  to  the  inclined  line  repre- 
senting "effective  depth"  and  from  the  intersection  of  this  line  run  down 
on  a  vertical  line  to  the  bottom  of  the  sheet  where  the  depth  of  slab  is 
read. 

II.  This  portion  of  the  chart  is  for  obtaining  the  area  of  steel  re- 
quired in  cinder  concrete  slabs.   Enter  the  diagram  at  the  bottom  at  the 
point  indicating  the  effective  depth  as  determined  by  I  and  run  up  verti- 
cally to  the  intersection  with  the  inclined  line.   From  this  intersection 
run  horizontally  to  the  left  hand  margin  of  the  sheet  where  the  total  area 
of  steel  required  per  foot  of  width  of  slab  is  read  from  the  scale. 


Cr  -  6  is  for  use  in  determining  the  number  of  round  steel  rods  re- 
quired to  make  up  a  required  area  of  steel.   The  diameters  of  the  rods  in 
inches  are  shown  at  the  bottom  of  the  sheet  and  by  following  the  operations 
indicated  by  the  arrows  on  the  chart  both  the  weight  and  area  of  the  rods 
can  be  obtained.   Cr  -  8a  is  self  explanatory. 


Cr  -  9  is  for  use  in  determining  the  number  of  square  steel  bars  re- 
quired to  make  up  a  required  area  of  steel.   The  thickness  of  the  bars  in 
inches  is  shown  at  the  bottom  of  the  sheet  and  by  following  the  operations 
indicated  on  the  chart  both  the  weight  and  area  of  the  bars  can  be  obtain- 
ed.  Cr  -  9a  is  self  explanatory. 


Cr  -  10  is  for  use  in  designing  round  columns  with  longitudinal  rein- 
forcements to  the  extent  of  not  less  than  one  per  cent  and  not  more  than 
four  per  cent  and  with  lateral  ties  of  not  less  than  1/4  in.  in  diameter, 
12  inches  apart  nor  more  than  16  diameters  of  the  longitudinal  bars.   The 
chart  is  used  as  follows: 

1.  Enter  the  right  hand  side  of  the  chart  using  the  total  load  to  be 
carried  by  the  column,  and,  as  indicated  by  the  arrow,  follow  horizontally 
across  to  the  inclined  line  indicating  the  per  cent  of  longitudinal  rein- 
forcement.  From  the  intersection  of  this  line,  as  indicated  by  arrow  No.  2, 
follow  the  vertical  line  to  the  bottom  of  the  sheet  -where  the  effective 
diameter  of  the  nolumn  is  shown. 


(7) 


II.  Using  the  effective  diameter  as  obtained  above,  enter  the  left 
h%and  side  of  the  chart  at  the  bottom  of  the  sheet  and  follow  the  vertical 
line  upwards  as  indicated  by  the  arrow  1,  to  the  intersection  vdth  the  in- 
clined line  indicating  th,e  same  percentage  of  steel  used  in  obtaining  the 
effective  diameter.  From  this  intersection  follow  the  horizontal  line  to 
the  left  hand  margin  where  the  total  area  of  longitudinal  steel  in  square 
inches  is  shown. 

EXAMPLE :  Design  column  for  load  of  200,000  Ibs.  Diameter  limited  to 
24  inches. 

Allowing  for  thickness  of  1  1/2  in.  for  fireproofing,  effective  dia- 
meter will  be  limited  to  21  inches. 

Entering  I  at  right  hand  margin  at  point  indicating  200,000  Ibs., 
run  over  to  intersection  with  the  vertical  line  representing  diameter  of 
21  in.  The  ratio  of  steel  area  to  area  of  concrete  is  found  to  be  .02. 

Entering  II  at  bottom  under  21  nttn  up  the  vertical  line  to  inter- 
section with  the  inclined  line  for  the  .02  ratio  of  steel  area  to  area  of 
concrete,  and  from  this  intersection  run  over  horizontally  to  the  left 
hand  margin  where  the  area  of  steel  is  found  to  be  6,9  sq.  in. 

From  Cr  -  8  and  Cr  -  9  the  required  size  and  number  of  rods  may  be 
figured. 

Using  1  in.  sq.  rods  (Cr  -  9)  the  area  of  each  rod  is  1  sq.  in.  and 
7  rods  are  required. 


Cr  -  11  is  for  use  in  designing  round  columns  reinforced  with  not 
less  than  one  per  cent  nor  more  than  four  per  cent  of  longitudinal  bars 
and  with  circular  hoops  or  spirals  not  less  than  one  per  cent  of  the 
volume  of  the  concrete  contained  within  the  reinforcement. 

In  using  this  diagram  the  diameter  of  the  column  and  area  of  the  lon- 
gitudinal steel  is  obtained  in  the  same  manner  as  in  Cr  -  10  and  in  addi- 
tion the  cross  section  of  the  hoops  is  obtained  by  the  portion  of  the  dia- 
gram in  the  lower  left  hand  corner,  marked  III,  as  follows:   Enter  the 
diagram  from  the  bottom  using  the  diameter  as  determined  by  I  and  follow 
the  line  vertically  upward  to  the  intersection  with  the  inclined  line  in- 
dicating the  approximate  predetermined  spacing  of  the  hoops.   From  the  in- 
tersection with  this  line  follow  the  horizontal  line  to  the  left  hand  mar- 
gin where  the  cross  sectional  area  of  the  hoops  is  at  once  read  off  in 
square  inches. 

Select  from  Cr-  8  or  Cr  -  9  the  size  rod  nearest  to  the  cross-sec- 
tion found  and  re-enter  the  diagram  on  the  left  hand  margin.  The  inter- 
section of  the  horizontal  line  indicating  cross-section  of  bar  selected 
and  the  vertical  line  indicating  the  effective  diameter  of  the  column 
gives  a  point  which  is  used  to  determine  by  interpolation  the  exact  spac- 
ing of  the  hoops. 

In  regard  to  the  spacing  of  the  hoops,  the  Joint  Committee  of  the* 
A.  S.  C.  E.  recommends  that  the  spacing  of  the  hooping  should  not  be 
more  than  1/6  of  the  enclosed  column  and  preferably  not  greater  than 
1/10  and  in  no  case  more  than  2  1/2  ip  . 


Cr  -  12  is  for  use  in  designing  square  columns  with  longitudinal 
reinforcements  to  the  extent  of  not  less  than  one  per  cent  and  not  more 
than  four  per  cent  and  with  lateral  ties  of  not  less  than  1/4  in,  in 
diameter  12  ins,  apart  nor  more  than  16  diameters  of  the  longitudinal 
bars.  This  chart  is  used  as  follows: 

I.  Enter  the  right  hand  side  of  the  chart  using  the  total  load 
to  be  carried  by  the  column  and,  as  indicated  by  the  arrow,  follow  hor- 
izontally across  to  the  inclined  line  indicating  the  per  cent  of  longi- 
tudinal reinforcement.  From  the  intersection  of  this  line,  as  indicated 
by  arrow  No.  2,  follow  the  vertical  line  to  the  bottom  of  the  sheet 
where  the  effective  dimensions  of  the  column  are  shown. 

II.  Using  the  effective  dimensions  as  obtained  above,  enter  the 
left  hand  side  of  the  chart  at  the  bottom  of  the  sheet  and  follow  the 
vertical  line  upwards  as  indicated  by  arrow  No.  1  to  the  intersection 
•with  the  inclined  line  indicating  the  same  percentage  of  steel  used  in 
obtaining  the  effective  dimensions.   From  this  intersection  follow  the 
horizontal  line  to  the  left  hand  margin  where  the  total  area  of  longi- 
tudinal steel  in  square  inches  is  shown. 


Cr  -  13  is  for  use  in  determining  variables  required  for  utiliz- 
ing the  slab  and  beam  formula.   The  upper  part  of  the  chart  shows  values 
of  j  as  varying  for  different  values  of  m  and  n. 

Entering  the  chart  at  the  bottom  with  the  value  of  m  determined 
for  the  particular  problem,  follow  the  vertical  line  to  its  intersection 
with  the  curved  line  representing  the  value  of  n,  and  from  this  inter- 
section follow  the  horizontal  line  to  the  left  hand  margin  where  the  val- 
ue of  j  is  shown. 

The  lower  part  of  the  chart  shows  values  of  k  for  varying  combina- 
tions of  n  and  p.  The  method  of  using  this  part  of  the  chart  is  similar 
to  the  above. 


Cr  -  14  is  also  for  use  in  determining  variables  for  use  in  the  con- 
crete beam  and  slab  formula.   The  upper  part  shows  values  of  m  as  depend- 
ing upon  different  values  of  n  and  p. 

The  lower  part  of  the  chart  shows  the  values  of  k  as  depending  upon 
m  and  n. 


Cr  -  15  and  Cr  -  16  are  for  determining  the  proportions  of  sand  and 
stone  per  unit  volume  of  cement  for  an  economic  mix. 

Cr  -  17  is  for  determining  the  amount  of  cement  required  and  is  for 
use  in  connection  with  Cr  -  16  and  Cr  -  15. 

The  following  notes  are  abstracted  from  an  article  describing  the 
theory,  method  of  computation  and  application  of  the  charts  to  field  prac- 
tice, published  in  Engineering  News,  April  20,  1905. 


(9) 


The  assumptions  of  the  theory  are  as  follows: 

(1)  The  voids  in  the  sand  should  be  filled  with  cement  paste. 

(2)  The  voids  in  the  stone  or  gravel  should  be  filled  with  mortar. 

(3)  There  should  be  a  small,  and  definite  excess  of  paste  over  the 
amount  necessary  to  fill  the  voids  in  the  sand. 

(4)  Likewise  there  should  be  a  small  and  definite  excess  of  mortar 
over  the  amount  necessary  to  fill  the  voids  in  the  stone. 

(5)  The  voids  in  the  sand  and  stone  are  easily  determinable.  And 
the  voids  of  the  stone  after  being  rammed  can  be  determined  before  addi- 
tion of  mortar  with  sufficient  accuracy  for  all  practical  purposes,  any 
error  here  being  on  the  safe  side. 

(-6)  The  excess  of  cement  serves  the  two-fold  purpose  of  compensat- 
ing for  irregularities  in  mixing,  preventing  accidental  voids,  and  of  sup- 
plying a  coat  of  cement  over  the  surface  of  each  grain  of  sand  and  piece 
of  stone.  A  largerexcess  of  cement  accomplishes  no  useful  object,  makes 
the  concrete  more  liable  to  cracks  and  checks,  and,  in  general,  reduces 
the  density  of  the  mass,  besides  adding  greatly  to  the  cost  of  the  work. 

The  extreme  stone  values  given  in  the  diagrams,  when  the  stone  voids 
are  35$  and  the  cement  voids  15$,  will  rarely  be  arrived  at  in  practice, 
but  the  diagrams  can  be  used  with  perfect  confidence  and  with  absolute  as- 
surance that  the  voids  will  be  filled  provided  that  the  voids  in  the  sand 
and  stone  are  properly  determined  in  the  first  instance.   It  will  be  noted 
that  the  voids  in  the  stone  are  supposed  to  be  the  same  in  the  first  deter- 
minations as  after  placing  in  the  mortar.  No  allowance  has  been  made  for 
consolidation  by  ramming.  This  does  not  mean  that  ramming  is  unnecessary. 

It  is  the  writer's  practice  to  mix  the  concrete  with  enough  water  to 
make  it  just  quake  under  ramming  and  then  to  ram  rapidly  and  briefly.  For 
watertight  work  slightly  more  water  is  added. 

Attention  is  called  to  the  long  sand  line.  Frequently  stone  is  not 
available  or  is  very  expensive,  when  gravel  is  cheap  and  plentiful.  By 
mixing  gravel  from  two  parts  of  the  bank,  the  voids  can  often  be  reduced 
to  15%  or  20$.  With  E  equal  to  10%  this  makes  splendid  concrete,  mixing 
in  the  cement  directly,  from  1  -  4-5  to  1  -  6,  and,  by  stretching  a 
point,  even  higher. 

It  will  be  noted  that  the  voids  in  sand  do  not  run  above  42$  in  the 
diagram.   Frequently  the  first  test  of  sand  will  show  45$  or  50$  voids, 
but  this  can  nearly  always  be  reduced  by  mixing  two  kinds  of  sand,  which 
is  more  economical  than  using  the  extra  amount  of  cement  involved. 

The  number  of  cubic  feet  of  cement  paste  which  can  be  obtained  from 
a  barrel  of  cement  depends  upon  a  number  of  factors,  the  thoroughness  of 
ramming,  the  voids  in  the  cement  itself,  amount  of  water  used,  etc.  The 
value  given,  3.5  cu.  ft.,  is  one  which  best  accords  with  the  writer's  ex- 
perience, and  is,  if  anything,  on  the  "safe  side." 

Classification  of  Concrete  for  Various  Uses 

For  purposes  of  estimation,  the  writer  classifies  concrete  by  the 
value  of  E,  and  in  writing  specifications  and  in  the  field  uses  the  ordi- 
nary nomenclature  of  sand  and  stone,  the  values  being  taken  from  the  dia- 
grams. 


(10) 


For  foundations  10$  concrete  is  recommended. 

For  abutments  and  piers,  15%  concrete  is  recommended. 

For  reinforced  work,  20$  concrete  is  recommended. 

For  thin  sections,  slabs  and  waterproof  work,  25%  concrete  is  recom- 
mended, with  an  excess  of  water. 

Test  for  Voids 

The  voids  in  the  stone  can  be  easily  obtained  by  anyone  in  the  field, 
but  the  sand  voids  are  difficult  to  get,  because  with  damp  sand  there  are 
air  bubbles  within  the  mass  that  prevent  the  entrance  of  water  into  all  the 
voids.  A  method  that  has  been  much  recommended  is  to  weigh  a  cubic  foot  of 
sand  and  figure  out  the  voids  by  means  of  the  specific  gravity  of  quartz, 
2.65,  or  165  Ibs.  per  ou.  ft. 

The  best  method  known  to  the  writer  is  as  follows:  Take  a  quart  meas- 
ure exactly  half  full  of  water,  and  pour  into  it  exactly  one  pint  of  the  dry 
sand.  If  the  water  rises  to  the  first  gill  mark  from  the  top  the  voids  are 
25$;  if  1  1/2  gill  marks  from  the  top,  voids  are  1.5-*-4  m  37.5$,  and  so  on. 

EXAMPLE  :  To  illustrate  the  use  of  the  diagrams.  Suppose  that  we  wish 
to  make  foundation  concrete,  and  find  the  voids  in  the  stone  and  sand  45$ 
and  Z5%  respectively.  Then,  assuming  E  equals  10$,  by  Cr  -  15  the  propor- 
tions by  volume  are  1:2*75:5.75. 

For  a  check,  the  voids  in  2.75  volumes  of  sand  are 

2.75  x  0.33  equals  0.91 

E  equals  10$  0.09 

Amount  of  cement  paste  required  1.00,  which 

checks. 

Also,  the  net  volume  of  sand  is  2.75  -  0.91  equals  1.84  volt. 
add  cement  paste  1.00 

Voids  in  stone  plus  10$  E  equals  2.84 

Therefore  voids  in  stone  equal  2.84-*-l.l  equals         2068 
Stone  equals  2.58-!-0.45  equals  5.75  vols.  which 

checks  with  the  diagram. 

We  wish  now  to  get  the  number  of  barrels  of  cement  per  ou.  yd.  of 
concrete.   On  Cr  -  17  for  E  equals  10$,  corresponding  to  the  5.75  ordi- 
nate,  we  have  the  abscissa  1.28,  or  practically  1.3,  Had  we  taken  E 
equals  20$,  with  the  same  voids  we  should  have  had  1:2.5:5  concrete  and 
1.40  bbls.  per  yard. 

The  amounts  of  stone  per  cu.  yd.  of  concrete  are  0.96  and  0.92, 
found  in  brackets  on  the  stone  line  of  the  diagrams. 

Take  the  data  of  the  first  example  and  suppose  there  is  a  gravel 
bank  available  not  more  than  1/4  mile  from  the  work.  We  can  get  screened 
gravel  delivered  for,  say  50  ots.  per  cu.  yd.  on  the  wagon.   Mix  1/4  ou. 
yd.  of  this  gravel  having  say  35$  voids,  with  every  yard  of  the  stone. 


(11) 


Then  in  every  cubic  yard  of  stone  there  will  be: 

Voids  0.45  Cu.  yd. 

Net  stone  0.55  " 

Gravel  0.25  " 

Voids  in  gravel  .25  x  .35  0.088  " 

Net  vol.  gravel  0.162  ' 

Gross  vol.  of  stone  and  gravel,  say          1.05 
Voids  in  stone  and  gravel,  1.05  -  0.71        0.34  " 

Voids  =  0.34-7-1.05  =  32. 4^,  or  say  35*i.   Our  mixture  with  the  same 
sand  then  becomes  1:2.75:7.4,  instead  of  1:2.75:5.75,  and  the  cement  is 
'reduced  from  1.3  bbls,  to  1  bbl.  per  yard  of  finished  concrete. 

The  cost  of  the  screened  gravel  has -been  »97-:-4  x  50  cts,  or  12.1 
cts.   The  extra  labor  of  mixing  will  on  the  most  liberal  estimate  not  be 
more  than  10  cts.  per  yard.  The  saving  in  cement  will  be  0.3  bbl.  per 
yard,  or,  if  the  cement  cost  $1.50  per  bbl.  on  the  work,  45  cts.  -  a  net 
saving  to  somebody  of  32.9  cts,  per  yard  of  concrete.  On  a  thousand  yard 
job  this  is  a  saving  of  $329,  at  an  expenditure,  on  the  part  of  the  engi- 
neer, of  an  hour's  time  and  a  little  gray  matter.  On  a  ten  thousand  yard 
job  the  saving  is  no  small  item,  and  even  on  a  small  piece  of  work  the 
saving  pays  the  cost  of  the  engineer's  time  many  fold.  Bear  in  mind  that 
the  mortar  is  just  as  rich  as  it  was  before,  and  that  the  resulting  con- 
crete should  be  just  as  strong.  These  figures  are  finely  drawn,  but  they 
are  well  within  the  limits  of  practice. 

The  diagrams  also  give  an  idea  of  some  of  the  pitfalls  attending  the 
use  of  our  old  standbys,  the  1:2:5  and  1:3:6  mixtures,  without  first  ascer- 
taining the  voids.  The  1:2:5  mixture  contains  an  unnecessarily  large 
amount  of  cement  in  the  mortar  and  too  much  stone  where  the  sand  voids  are 
Z4%  or  less.  The  1:3:6  mixture  is  all  right  for  30$  sand  and  47#  stone, 
but  for  Z5%  stone  8  parts  may  be  used,  E  being  10$. 


Cr  -  18  shows  the  comparative  strengths  of  different  mixtures  of  con- 
crete as  recommended  by  the  Joint  Committee  of  the  A.  S.  C.  E.   The  combi- 
nation of  the  volume  of  fine  and  coarse  aggregates  measured  separately  for 
a  given  mixture  of  concrete  is  the  abscissa  and  the  limit  of  strength  in 
pounds  is  shown  by  the  ordinates.  At  the  right  hand  side  of  the  chart  are 
shown  the  values  of  n  as  recommended  by  the  Joint  Committee  of  the  A.  S.  C. 
E.  for  different  strengths  of  concrete. 


Cr  -  19  is  for  determining  the  cost  per  ou.  yd.  of  materials  for  con- 
crete of  various  mixes. 

I.  By  following  steps  indicated  by  arrows  the  volume  of  batch  con- 
taining 1  bbl.  of  cement  is  obtained. 

II.  By  following  steps  indicated  by  arrows  the  cost  of  sand  and 
aggregate  per  batch  containing  1  bbl.  of  cement  is  obtained. 

III.  From  point  indicating  volume  of  batch  found  from  I.   Follow 
vertical  line  to  intersect  horizontal  line  through  the  point  giving  cost 
of  cement,  sand  and  aggregate.   (Value  from  II.  plus  cost  of  1  bbl.  of  ce- 
ment.) From  this  intersection  follow  inclined  line  indicating  Cost  of 


Materials  per  cu.  yd.  to  the  vertical  line  "Cost  of  Cement,  Sand  and  Aggre- 
gate per  Cu.  Yd."  where  the  cost  per  cu.  yd.  is  at  once  read.   The  cost  per 
cu.  yd.  may  also  be  read  by  interpolation  of  the  values  shown  by  the  lines 
showing  "Cost  of  Materials  per  Cu.  Yd."  between  which  the  intersection  of 
the  vertical  line  (from  the  "Vol.  of  Batch  Containing  1  bbl.  of  Cement")  and 
the  horizontal  line  (From  the  "Cost  of  Cement,  Sand  and  Agg.  per  Batch  Con- 
taining 1  bbl.")  falls. 

EXAMPLE  :  What  will  be  the  cost  of  1  cu.  yd.  of  1:3:6  concrete  with  the 
average  of  voids  in  sand  and  stone  40$,  cost  of  sand  $2.25  per  cu.  yd.,  stone 
$1,75  per  cu.  yd.,  cement  $2.00  per  bbl.? 

I.  Enter  the  left  hand  margin  of  I  at  9  =  (3  +  6).  Follow  horizontal 
line  to  right  to  intersection  with  inclined  line  indicating  average  voids  of 
4C$.   From  this  intersection  follow  vertical  line  to  bottom  of  diagram  I 
where  volume  of  batch  is  shown  to  be  0.9  cu.  yd. 

II.  Enter  the  right  hand  margin  of  II  at  9  and  follow  horizontal  line 
to  left  to  intersection  with  inclined  line  $2.00,  (the  average  price  of  sand 
and  aggregate  per  cu.  yd.).  From  this  intersection  follow  vertical  line  to 
bottom  of  diagram  II  where  the  cost  of  sand  and  aggregate  per  batch  contain- 
ing 1  bbl.  of  cement  is  shown  to  be  $2.50. 

III.  Enter  top  of  diagram  III  at  0.9  and  follow  the  vertical  line  down 
to  the  intersection  with  the  horizontal  line  from  a  point  representing  the 
"Cost  of  Cement,  Sand  and  Agg.  per  Batch  Containing  1  bbl.",  i.e.  $4.  50 
($2.50-*-$2.00).  The  cost  per  cu.  yd.  is  found  to  be  $5.00. 

FORMULAS  FOR  REINFORCED  CONCRETE  CONSTRUCTION* 
1.   STANDARD  NOTATION 


(a)  Rectangular  Beams.  v 

The  following  notation  is  recommended:      °e-f*AKTl^f       °F  CAL"*OI?MI 

fs  »  tensile  unit  stress  in  steel;  -&KELEY.         iNe 

fc  •  compreesive  unit  stress  in  concrete; 

Es  =  modulus  of  elasticity  of  steel; 

Ec  a  modulus  of  elasticity  of  concrete; 


n  .  s  ; 
~Ec 

M  *  moment  of  resistance,  or  bending  moment  in  general; 

AS  *  steel  area; 

b  -  breadth  of  beam; 

d  -  depth  of  beam  to  center  of  steel; 

k  =  ratio  of  depth  of  neutral  axis  to  depth,  d; 

z  -  depth  below  top  to  resultant  of  the  compressive  stresses; 

3  =  ratio  of  lever  arm  of  resisting  couple  to  depth,  d; 

jd  -  d  -  z  •  arm  of  resisting  couple; 

P  =  steel  ratio  •  A_  • 

bd 
(b)   T-Beams. 

b(  -  width  of  flange; 

b  •  width  of  stem; 

t  =  thickness  of  flange. 

*  Suggested  in  1917  by  Committee  of  A.  S.  C.  E. 

(13) 


(c)  Beams  Reinforced  for  Compression. 


A1  s  area  of  compressive  steel: 

p1  •  steel  ratio  for  corapressive  steel: 

fs'-  compressive  unit  stress  in  steel; 

C  =  total  compressive  stress  in  concrete; 

C1  •  total  compressive  stress  in  steel; 

d1  «  depth  to  center  of  oorapressive  steel; 

z  s  depth  to  resultant  of  C  and  C1. 

(d)  Shear,  Bond  and  Web  Reinforcement. 


V 

V1 

v 

u 

o 


T 
s 


total  shear; 

total  shear  producing  stress  in  reinforcement; 

shearing  unit  stress; 

bond  stress  per  unit  area  of  bar; 

circumference  of  perimeter  of  bar; 

sum  of  the  perimeters  of  all  bars; 

total  stress  in  single  reinforcing  nember; 

horizontal  spacing  of  reinforcing  members. 


(e)  Columns. 


A  =  total  net  area; 

Ag  •  area  of  longitudinal  steel; 

AQ  a  area  of  concrete; 

P  a  total  safe  load. 


2.   FORMULAS. 
(a)  Rectangular  Beams. 


Position  of  neutral  axis,  k  s  y  2pn  +  (pn)  - 


pn. 


Arm  of  resisting  couple,  j  -  1  -  i  k (2) 

3 

For  fg  a  15000  to  16000  and  f0  s  600  to  650,  j  may  be  taken  at 


Fiber  Stresses,  fs  •  Asjd 

f     ZU 

Jc  *     f 

jkbd' 


^ 

&  J 

-.(3) 
-.(4) 


Steel  ratio,  for  balanced  reinforcement,  p  •  ^ 

2 


(e)     Columns. 

Total   safe   load,    P  .  fc(Ac+nAs)   =    F.At  |_1+  (n 

8 
Unit        stresses,    fc  «  _  P  __ 


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Engineering 


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